Are every two separated nets in the plane bilipschitz equivalent? In the late 1990s, Burago and Kleiner and, independently, McMullen resolved this beautiful question negatively. Both solutions are based on a construction of a density function that is not realizable as the Jacobian determinant of a bilipschitz map. McMullen’s construction is simpler than the Burago–Kleiner one, and we provide a ...

We study the canonical metric on a compact Riemann surface of genus at least two. This natural metric is the pullback, via the period map, from the Euclidean metric on the Jacobian variety of the surface. While it is known that the canonical metric is of nonpositive curvature, we show that its Gaussian curvatures are not bounded away from zero nor negative infinity when the surface is close to ...

In this paper, we describe a new method to map a texture on a surface with a spatially-variant filter. Our filter takes into consideration the effects of anisotropy using a Jacobian approximation while computing the sampling rate, and the interpolation weights are computed with a sinc function. We also discuss how to do forward and backward mapping with the filter, and extend our algorithms to ...

This paper presents a new approach to inverse consistent image registration. A uni-directional algorithm is developed using symmetric cost functionals and regularizers. Instead of enforcing inverse consistency using an additional penalty that penalizes inconsistency error, the new algorithm directly models the backward mapping by inverting the forward mapping. The resulting minimization problem...

A supersymmetric generalization of the Krichever map is used to construct algebro-geometric solutions to the various super KadomtsevPetviashvili (SKP) hierarchies. The geometric data required consist of a suitable algebraic supercurve of genus g (generally no t a super Riemann surface) with a distinguished point and local coordinates (z, 0) there, and a generic line bundle of degree g 1 with a ...

Introduction and Review 2 1. Period domains and monodromy 4 2. Elliptic curves 8 3. Period mappings: an example 12 3.1. Asymptotics of the period map 14 4. Hodge structures of weight one 15 5. Hodge structures of weight two 18 6. Poincaré residues 20 7. Properties of the period mapping 24 8. The Jacobian ideal and the local Torelli theorem 25 9. The Horizontal Distribution Distance Decreasing P...

This paper is to present a geometrical proof of the plane Jacobian conjecture for rational polynomials by an approach of Newton-Puiseux data and geometry of rational surfaces. The obtained result shows that a polynomial map F = (P, Q) : C −→ C with PxQy − PyQx ≡ const. 6= 0 has a polynomial inverse if the component P is a rational polynomial, i.e. if the generic fiber of P is the 2-dimensional ...

Two proper polynomial maps f1, f2 : C 2 −→ C are said to be equivalent if there exist Φ1, Φ2 ∈ Aut(C) such that f2 = Φ2 ◦ f1 ◦ Φ1. We investigate proper polynomial maps of topological degree d ≥ 2 up to equivalence. Under the further assumption that the maps are Galois coverings we also provide the complete description of equivalence classes. This widely extends previous results obtained by Lam...

In this paper, three key theorems (the open mapping theorem, the inverse function theorem, and the implicit function theorem) for continuously differentiable maps are shown to hold for nonsmooth continuous maps which are not necessarily Lipschitz continuous. The significance of these extensions is that they are given using generalized Jacobians, called approximate Jacobians. The approximate Jac...

Let ~ C R n and n > 4 be even. We show that if a sequence {uJ } in W l'n/2 (f2; R n) is almost conformal in the sense that dist (Vu j , R+ SO(n) ) converges strongly to 0 in L n/2 and i f u j converges weakly to u in W 1,n/2, then u is conformal and VuJ ~ Vu strongly in Lqoc for all 1 < q < n/2. It is known that this conclusion fails i f n~2 is replaced by any smaller exponent p. We also prove ...